How to snap part of a mesh to the grid?
You can CtrlAltQ to toggle Quad View, then click to the nearest grid crossing to place 3D Cursor there, then ShiftS > Cursor to Grid to snap the 3D Cursor in two of the side views (to snap it precisely to the grid dimension as you want). Then, ShiftS > Selection to Cursor (Offset) to offset the selection to the 3D Curor, without collapsing:



NOTE: Snapping cursor to grid currently depends much on the view zooming, in other words, the visual grid scale, so, if it cannot be easily snapped to the crossing, just give it another try after zooming in or out.



UPDATE: For your concern in the comment, you can temporarily add an empty to assist snapping: